Inertia groups of a toric DM stack, fake weighted projective spaces, and labelled sheared simplices

TL;DR

This paper explicitly computes inertia groups of points in toric Deligne-Mumford stacks, linking geometric and combinatorial data, and characterizes when these stacks are equivalent to weighted or fake weighted projective stacks.

Contribution

It provides a detailed computation of inertia groups for toric DM stacks and characterizes stacks that are global quotients or equivalent to weighted projective stacks.

Findings

Explicit formulas for inertia groups in terms of stacky fans.

Characterization of stacks as weighted or fake weighted projective stacks.

Detailed example computations for labelled sheared simplices.

Abstract

This paper determines the inertia groups (isotropy groups) of the points of a toric Deligne-Mumford stack [Z/G] (considered over the category of smooth manifolds) that is realized from a quotient construction using a stacky fan or stacky polytope. The computation provides an explicit correspondence between certain geometric and combinatorial data. In particular, we obtain a computation of the connected component of the identity element $G_0 \subset G$ and the component group $G/G_0$ in terms of the underlying stacky fan, enabling us to characterize the toric DM stacks which are global quotients. As another application, we obtain a characterization of those stacky polytopes that yield stacks equivalent to weighted projective stacks and, more generally, to `fake' weighted projective stacks. Finally, we illustrate our results in detail in the special case of labelled sheared simplices, where explicit computations can be made in terms of the facet labels.